Abstract: In quantum teleportation, the state of a single quantum system is disembodied
into classical information and purely quantum correlations, to be later
reconstructed onto a second system that has never directly interacted with the
first one. This counterintuitive phenomenon is a cornerstone of quantum
information science due to its essential role in several important tasks such
as the long-distance transmission of quantum information using quantum
repeaters. In this context, a challenge of paramount importance is the
distribution of entanglement between remote nodes, and to use this entanglement
as a resource for long-distance light-to-matter quantum teleportation. Here we
demonstrate quantum teleportation of the polarization state of a
telecom-wavelength photon onto the state of a solid-state quantum memory.
Entanglement is established between a rare-earth-ion doped crystal storing a
single photon that is polarization-entangled with a flying telecom-wavelength
photon. The latter is jointly measured with another flying qubit carrying the
polarization state to be teleported, which heralds the teleportation. The
fidelity of the polarization state of the photon retrieved from the memory is
shown to be greater than the maximum fidelity achievable without entanglement,
even when the combined distances travelled by the two flying qubits is 25 km of
standard optical fibre. This light-to-matter teleportation channel paves the
way towards long-distance implementations of quantum networks with solid-state
quantum memories.